The Space of Closed $G_2$-Structures. I. Connections

Abstract

In this article, we develop foundational theory for geometries of the space of closed $G_2$-structures in a given cohomology class as an infinite-dimensional manifold. We introduce Sobolev-type metrics, construct their Levi-Civita connections, formulate geodesic equations, and analyse the variational structures of torsion free $G_2$-structures under these Sobolev-type metrics.

arXiv comment51 pages

References in corpus (16)

• Flows of G_2 Structures, I
• Laplacian flow for closed G_2 structures: Shi-type estimates, uniqueness and compactness
• Laplacian Flow for Closed $G_2$-Structures: Short Time Behavior
• Stability of torsion-free G_2 structures along the Laplacian flow
• Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps
• Hypersymplectic 4-manifolds, the $G_2$-Laplacian flow and extension assuming bounded scalar curvature
• Laplacian flow for closed G_2 structures: real analyticity
• Moduli spaces of G2 manifolds
• Local geometry of the G2 moduli space
• A gradient flow of isometric ${\mathrm{G}}_2$ structures
• Estimates and monotonicity for a heat flow of isometric G2-structures
• A flow of isometric $G_2$-structures. Short-time existence
• Regularity scales and convergence of the Calabi flow
• The Calabi's metric for the space of Kaehler metrics
• Geodesics in the space of Kahler cone metrics, II. Uniqueness of constant scalar curvature Kahler cone metrics
• The Dirichlet and the weighted metrics for the space of Kahler metrics